Non-uniqueness of infinitesimally weakly non-decreasable extremal dilatations

Abstract

In this paper, it is shown that a weakly non-decreasable dilatation in an infinitesimal Teichm\"uller equivalence class can be not a non-decreasable one. As an application, we prove that if an infinitesimal equivalence class contains more than one extremal dilatation, then it contains infinitely many weakly non-decreasable extremal dilatations.